6 2 Motivation In mathematics, Eq. (2.1) is termed a fi st-order ordinary differential equation and, as it requires an initial value for C,itisalsoreferredtoasaninitial-value problem. 2.1.3 Example Grandma used to keep a 10-litre carton of red wine in a storage room. I could not resist and crept every night to this room to get myself 1 litre of wine. To avoid grandma noticing the fast disappearance of the wine, each night, I topped up the car- ton with tab water. However, I was caught soon or do you really think that grandma would not notice changes in the wine’s taste and color owing to dilution with water? Anyway, the reader can easily work out how the concentration of wine changed with every day of my early-year drinking habit. All that needs to be done is to take away 10% of the wine concentration on a daily basis. The following table shows the result of this. Day Wine content (l) Wine concentration (%) 0 10 100 19.090.0 28.181.0 37.29 72.9 46.56 65.6 55.959.0 65.31 53.1 74.78 47.8 84.343.0 93.87 38.7 10 3.49 34.9 11 3.14 31.4 12 2.82 28.2 13 2.54 25.4 14 2.29 22.9 15 2.06 20.6 16 1.85 18.5 17 1.67 16.7 18 1.515.0 19 1.35 13.5 20 1.22 12.2 21 1.09 10.9 22 0. 98 9.8 Figure 2.1 shows a graph of the outcome of this f rst numerical prediction model, only using a piece of paper and a pen. As expected, there is a decrease of wine content as the days go past. In fact, this decrease is approximately exponential. 2.1 The Decay Problem 7 Fig. 2.1 Evolution of wine content 2.1.4 How to Produce a Simple Graph with SciLab This exercise explains how to use SciLab to produce Fig. 2.1. Step1: Open a suitable text editor and insert and save the three rows of the above table (without header) in a f le called, for instance, “winethief.txt”. Step 2: Fire up SciLab and change to the folder containing this file Step 3: Enter the following commands (each followed by <return>): x=read(“winethief.txt”,-1,3); plot(x(:,1),x(:,2)); xtitle(“ ”,“Time (days)”, “Wine content (litres)”,“ ”) The “read” command imports data from the file The parameter “−1” is used if the number of rows is unknown. “3” means: read the f rst three columns, “−1” does not work for columns. The above plot command plots values of row 2 against those of row 1. The last line adds axis labels. The semicolon “;” suppresses output to the SciLab command window. Step 4: Change line width, font size, etc. in the graphics window by selecting “Edit Figure Properties” in the “File” menu or by clicking “GED” in the graphics window. The reader is encouraged to “play around” with all options available. Step 5: Export the graph using a suitable format. I often selected the PostScript format and converted images into Portable Network Graphics file (PNG) with ImageMagick. 8 2 Motivation 2.2 First Steps with Finite Differences 2.2.1 Finite Time Step and Time Level With the use of a discrete time step Δt, we may formulate (2.1) as: C n+1 −C n Δt =−κC n (2.2) where the integer n refers to a certain time level. This time index must not be con- fused with “to the power of”. Conventionally, n = 0 gives the concentration at start time of your simulation, n = 1 refers to the concentration after one time step (of Δt in duration), n = 2 refers to the concentration after two time steps, and so on. 2.2.2 Explicit Time-Forward Iteration It is convenient to move the unknown variable in (2.2) to the left-hand side of the equation and shuffl all known terms to the right-hand side. This gives: C n+1 = C n −Δt ·κ ·C n = ( 1 −Δt ·κ ) C n (2.3) where C n=0 refers to the initial concentration that needs to be prescribed together with values of κ and Δt. This iterative method uses values known at a certain time level n to predict the value of C at the next time level n + 1 and is therefore called explicit time-forward iteration. 2.2.3 Condition of Numerical Stability for Explicit Scheme As can be seen from (2.3), with every time step, the concentration becomes dec- imated by a certain fraction in an iterative manner. This fraction is given by the product κ · Δt. It is at hand to request that this product be less than unity, other- wise the predicted concentration would become negative, which would not make sense. For κ · Δt > 2, the magnitude of concentration would even increase. The corresponding condition: Δt < 1 κ (2.4) is called a condition of numerical stability. Hence, the prediction of (2.3) is only sta- ble when (2.4) is satisfied Accordingly, the maximum time step that can be chosen depends on the value of κ. 2.2 First Steps with Finite Differences 9 2.2.4 Implicit Time-Forward Iteration Alternatively, Eq. (2.1) can be formulated in finite-di ference form as: C n+1 −C n Δt =−κ ·C n+1 (2.5) where the concentration on the right-hand side is evaluated at the next time level n +1. This approach might sound strange to some readers, but if we reorganize this equation, we yield a clear separation of known and unknown terms of the form: C n+1 = C n ( 1 +Δt ·κ ) (2.6) The clear advantage of this implicit scheme over the explicit approach is that it is numerically stable for any value of Δt. The denominator in the later equation always exceeds unity, so that concentration gradually decreases with time (and never changes sign). 2.2.5 Hybrid Schemes One could also use a mix between the explicit and the implicit scheme, which can be formulated as: C n+1 −C n Δt =−α ·κ ·C n+1 − ( 1 −α ) κ ·C n (2.7) where the weighting factor α (the Greek symbol “alpha”) has to be chosen from a range between zero and unity. The choice of α = 1 gives the fully implicit scheme, whereas α = 0 leads to the fully explicit scheme. With α = 0.5, we obtain a so-called semi-implicit scheme. 2.2.6 Other Schemes There are more advanced schemes such as the “Runge-Kutta scheme” or the “Adams-Bashforth scheme”, not discussed here, that in addition to current and future time level consider a number of sub-time steps. The accuracy and efficien y of the prediction model can be significantl improved with such schemes. 10 2 Motivation 2.2.7 Condition of Consistency The exact analytical solution of the decay problem (2.1) for an initial concentration of C o is given by: C(t) = C o exp(−κ ·t) (2.8) where “exp” is the exponential function. A numerical model is said to be consistent if its finite-di ference solution converges toward the solution of the governing dif- ferential equation when the numerical time step (or grid size) is made vanishingly small. This implies that the concentration predicted by our model should get the closer to the true solution for a decrease of the time step Δt. 2.2.8 Condition of Accuracy A certain error referred to as truncation error is made when using finit differences. Round-off errors are another source of error, being related to the fact that computers can represent numbers only with a finit number of digits. Both errors should stay reasonably small over the duration of a simulation. 2.2.9 Condition of Eff ciency Large programs may require substantial computer space for data output and storage, and completion of model runs may take a long time. Hence, model codes have to be written in an efficien manner such that the task is completed within a reasonable time span and without “stuffin up” the computer with enormous amounts of data. 2.2.10 How Model Codes Work The compiler translates the FORTRAN 95 code line by line and from top to bottom. This implies that parameters must be declared and specifie before they can be manipulated. Declaration means specificatio of the type of the parameter. There are integers, real numbers, arrays, characters and logical parameters. Specificatio means allocation of values to parameters. In principle, each line of a computer code can only have a single unknown on the left-hand side of an equation, such as “x = b +c”, where b and c have to be declared and assigned values farther up in the code, and x has to be at least declared. 2.2 First Steps with Finite Differences 11 2.2.11 The First FORTRAN Code Write a f rst FORTRAN code that prints “Hello World” on the display. This is an old tradition among modellers. The solution is the code: PROGRAM firs write(6, ∗ )“Hello World” END PROGRAM firs FORTRAN programs start with a PROGRAM name statement and finis with a END PROGRAM name statement. My program is called “first” but the reader is free to choose a different name. Save this fil as “first.f95” 2.2.12 How to Compile and Run FORTRAN Codes Open the Command Prompt window (on Windows systems this is found under Start => All Programs => Accessories => Command Prompt) and move to the folder containing your FORTRAN source file Step 1: Compile the program by entering the command: g95 -o first. xe first.f9 where “-o” specifie the name of the executable program. Step 2: Correct errors until the compiler does not return any error messages. Step 3: If the compiling process was successful, the newly created fil “first. xe” can be executed by simply double-clicking its icon in a f le window or by entering “first <return> in the Command Prompt window. The result of this will be the display of “Hello World” in the Command Prompt window. Congratulations! 2.2.13 A Quick Start to FORTRAN All constants, parameters, variables and arrays have to declared before use. Hereby, full numbers called “integer” (−3, 0, 1, 3, etc.) are distinguished from decimal num- bers called “real” (1.2, 4.2, −5.23, etc.). Other options are logical expressions (true or false) and text (characters). Constant parameters are declared at the beginning of the code with: INTEGER, PARAMETER :: nx = 11 ! horizontal dimension INTEGER, PARAMETER :: nz = 5 ! vertical dimension 12 2 Motivation REAL, PARAMETER :: G = 9.81 ! acceleration due to gravity REAL, PARAMETER :: RHOREF = 1028.0 ! reference density REAL, PARAMETER :: PI = 3.14159265359 ! pi Text after a pronunciation mark is treated as a comment and ignored by the com- piler. Although not required, comments are very useful aids to highlight the structure of the program and as reminders for future uses. Parameters that are allowed to change values during the program’s execution are declared with: REAL :: wspeed ! wind speed INTEGER :: k ! grid index CHARACTER(3) :: txt In this example, “txt” is a character with three letters. One-dimensional and two- dimensional arrays are declared with: REAL :: eta(0:nx+1) ! sea-level elevation REAL :: w(0:nz+1,0:nx+1) ! vertical velocity With nx=11 and nz =5, for instance, “eta” obtains 11+2=13 so-far unassigned elements: eta(0), eta(1), ··· , eta(11), eta(12), and “w” is a two-dimensional array of 13 columns and 7 rows. After the declaration section, values can be assigned to these arrays using DO loops such as: DO k = 0,nx+1 IF(k > 50) THEN eta(k) = 1.0 ELSE eta(k) = 0.0 END IF END DO This DO-loop repeats certain calculations for the index “k” running from 0 at steps of 1 to nx+1. If the reader wants to do it the other way around, the solution is: DO k = nx+1,0,−1 IF(k > 50) THEN eta(k) = 1.0 ELSE eta(k) = 0.0 END IF END DO This example also includes an IF statement. Options are “>” (greater than), “<” (less than), “==” (equal to), “>=” (greater or equal), “<=” (less or equal), and “/=” 2.3 Exercise 1: The Decay Problem 13 (not equal). If there is only one line in an IF-statement, the “ELSE” and closing “END IF” statements can be dropped, such as in the following example: DO k = nx+1,0,−1 eta(k) = 0.0 IF(k > 50) eta(k) = 1.0 END DO Files for output can be opened with the statement: OPEN(10, fil = ‘Ex1a.txt’, form = ‘formatted’, status = ‘unknown’) The f rst entry (10) is a reference unit number for subsequent WRITE or READ statements. The “file entry specifie the desired filename The “form” entry spec- ifie whether to have ASCII numbers or binary numbers as output. I chose ASCII output. The status entry “unknown” implies new creation of a fil if this does not exist, otherwise an existing f le will be overwritten. Be careful not to overwrite file that might be needed in the future. Other status options are “new” or “old”. Output of data is done via “WRITE” statements such as WRITE(10, ∗ )G where the unit number (10 here) refers to a f le opened before, and the “*” symbol creates a standard format. Note that the unit number 6 is reserved for output to the screen as in our firs FORTRAN code. Similarly, “READ(5,*)” reads input from the keyboard. Several outputs at once can be produced with: WRITE(10, ∗ ) eta(10), eta(20), eta(30) Doing this repeatedly, there will be three columns of data. Output of an entire row of an array is done with: WRITE(10, ∗ ) (eta(k), k = 1,nx) Files no longer needed for output should be closed with the statement: CLOSE(10) 2.3 Exercise 1: The Decay Problem 2.3.1 Aim The aim of this exercise is to predict the decay of a substance according to (2.1) using a FORTRAN code based on either the explicit or the implicit scheme. 14 2 Motivation 2.3.2 Task Description Consider a substance that has an initial concentration of 100% and use a decay constant of κ = 0.0001 per second (or κ = 10 −4 s −1 ). Choose different values of the time step to verify whether the prediction becomes more accurate for a fine temporal resolution. Explore both the explicit and the implicit scheme. 2.3.3 Instructions Use any text editor to write the FORTRAN code and save the f le under the name “Exercise1.f95”. Blanks or other unusual symbols are not permitted here. Other filename may be used, but the reader should make sure that the f lename is not too long and that it has something to do with the exercise. The f le extension “f95” identifie this fil as a FORTRAN 95 source code. 2.3.4 Sample Code The Fortran code for this exercise, called “winethief.f95” can be found in the folder “Exercise 1” on the CD-ROM accompanying this book. 2.3.5 Results As a result of the model run, the data output file “output1.txt” or “output2.txt” should appear in the f le list. The MODE parameter in the code switches between the explicit and the implicit schemes. To avoid the recompiling procedure, values for “mode” could be alternatively read from the keyboard with “READ(5,*) mode”. Figure 2.2 shows model results for a time step of Δt = 3600 s using either the explicit scheme (2.3) or the implicit scheme (2.6). As can be seen, the explicit scheme slightly underestimates the correct concentration, whereas the implicit scheme slightly overestimates concentration. A semi-implicit approach would prob- ably give the best solution, but this remains to be verifie by the reader. With a time step of 3600 s, completion of the model run took only a few seconds on my computer. The accuracy of the prediction can be substantially improved with choice of a much fine temporal resolution with a time step of, say, Δt = 1s , which the reader can easily verify. 2.3.6 Additional Exercise for the Reader Repeat this exercise with use of the hybrid scheme (2.7) and explore the solutions for α = 0.25, 0.5 and 0.75. 2.4 Detection and Elimination of Errors 15 Fig. 2.2 Evolution of concentration (%) as a function of time. The upper and lower curves show results using the implicit and the explicit schemes, respectively. The middle curve displays the analytical solution according to Eq. 2.8 2.4 Detection and Elimination of Errors 2.4.1 Error Messages If a FORTRAN code contains errors, the compiler will return one or more error messages. There are a few important steps to follow for successful detection and elimination of errors. 2.4.2 Correct Errors One by One Only correct one error at a time with reference to the f rst error message. Often other errors are just followers of the f rst one. Similarly, an important rule is that the code should be compiled after each single alteration made. It can be tedious to locate errors after a dozen changes have been implemented without verification Errors are often the result of a lack of concentration. 2.4.3 Ignore Error Message Text Occasionally, the compiler’s error messages are confusing and misleading. There- fore, I recommend to ignore message text and rather focus on the line number this message refers to. [...]... Fig 3 .2 The Cartesian coordinate system D1 = 2 2 2 x1 + y1 + z 1 The nice thing about vectors is that they can easily be appended by adding up the individual vector components For instance, the sum of the location vectors (0, 5 m, −3 m) and ( 2 m, −5 m, 2 m) gives a new location vector ( 2 m, 0, −5 m) The distance between two locations (x1 , y1 , z 1 ) and (x2 , y2 , z 2 ) is given by: D2−1 = (x2 −... certain direction J K¨ mpf, Ocean Modelling for Beginners, a DOI 10.1007/978-3-6 42- 00 820 -7 3, C Springer-Verlag Berlin Heidelberg 20 09 17 18 3 Basics of Geophysical Fluid Dynamics 3 .2. 2 Contours and Contour Interval If a scalar varies spatially, this implies that it exhibits certain direction-dependent gradients Gradients in the spatial distribution of a scalar therefore constitutes a vector field To avoid... and (x2 , y2 , z 2 ) is given by: D2−1 = (x2 − x1 )2 + (y2 − y1 )2 + (z 2 − z 1 )2 In summary, location is specifie by a vector that points from the point-of-origin to a certain spot in space The magnitude of this vector is distance 3.3 .2 Calculation of Distances with SciLab In the following example, we calculate the distance of the location (4.1 m, 2. 6 m, −5.3 m) from the point-of-origin Start SciLab... in component form in the x-, y- and z-directions For convenience, we write this as (u, v, w) Since velocity is change of location of a moving parcel with time, we can also write: 20 3 Basics of Geophysical Fluid Dynamics dx∗ dt dy ∗ v= dt dz ∗ w= dt u= (3.1) where (x ∗ , y ∗ , z ∗ ) is the location of a parcel Speed is the magnitude of velocity and is given by: Speed = u 2 + v 2 + w2 For example (u,... 10 2 = 0.01 kilo (k) = 103 = 1000 3 .2 Scalars and Vectors 3 .2. 1 Difference Between Scalars and Vectors A scalar is a physical quantity without directional information Temperature and pressure are examples of this A vector, on the other hand, carries information of both magnitude and direction Velocity is an example of this A car moves at a certain speed into a certain direction J K¨ mpf, Ocean Modelling. .. negative y-direction; whereas (u,v,w) = (2 m/s, 2 m/s, 0.0) √ describes movement at a speed of 8 ≈ 2. 83 m/s diagonally across the x-y plane 3.4 Types of Motion 3.4.1 Steady-State Motions A steady state is a situation in which currents do not show any time variations This implies that there is a balance between all forces involved Considerations of steady-state force balances are useful tools in geophysical... Sinusoidal Waveform It is convenient to use the sinusoidal function to describe waves in a mathematical manner This function is based on radians and a complete cycle relates to a change of 3.5 Visualisation of a Wave Using SciLab 21 its argument by 2 (the Greek symbol “pi”), where π is about 3.1415 Accordingly, we can express a wave travelling in the x-direction as: A(x, t) = Ao sin 2 x t − λ T (3 .2) where... moves is called phase speed of a wave 3.5 .2 Sample Script The SciLab script for this wave demonstration, called “WaveSim.sce”, can be found in the folder “Miscellaneous” on the CD-ROM of this book Before using this script, however, the reader should read the following brief introduction to SciLab scripting Fig 3.3 Snapshot of organised wave motions by bars 22 3 Basics of Geophysical Fluid Dynamics 3.5.3... should take a good rest, perhaps a walk or a sleep Breaks are always important! 2. 4.6 Display Warnings Warnings can be displayed with addition of “-Wall” (display all warnings) in the compiling command For Exercise 1, for example, this command reads: g95 -Wall -o winethief.exe winethief.f95 Warning messages should be explored for potential errors in the code Chapter 3 Basics of Geophysical Fluid Dynamics... (kg) For convenience, temperatures are expressed in degrees Celsius (◦ C), which is the thermodynamic temperature in Kelvin (K) plus a constant of 27 3.16 Time lapse is translated into more convenient units such as minutes, hours, days, weeks, months, years, etc Other units can be derived from the above base units For example, volume is expressed in cubic metres (m3 ) Often, we also use symbols for multipliers . (%) 0 10 100 19.090.0 28 .181.0 37 .29 72. 9 46.56 65.6 55.959.0 65.31 53.1 74.78 47.8 84.343.0 93.87 38.7 10 3.49 34.9 11 3.14 31.4 12 2. 82 28 .2 13 2. 54 25 .4 14 2. 29 22 .9 15 2. 06 20 .6 16 1.85 18.5 17. (−2m, −5m, 2 m) gives a new location vector (−2m, 0, −5m). The distance between two locations (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) is given by: D 2 1 =  (x 2 − x 1 ) 2 +(y 2 − y 1 ) 2 +(z 2 −. direction. J. K ¨ ampf, Ocean Modelling for Beginners, DOI 10.1007/978-3-6 42- 00 820 -7 3, C  Springer-Verlag Berlin Heidelberg 20 09 17 18 3 Basics of Geophysical Fluid Dynamics 3 .2. 2 Contours and Contour